Pocket Games
The 15 puzzle
The 15 puzzle dates from 1880 and it has played the same trick on everyone ever since: the first ten tiles slot in almost by themselves, the last two look impossible. They are not — you just have to stop improvising and follow an order.
The rules
A grid of numbered tiles and one empty square. You can only move a tile adjacent to the gap, which really means: the gap moves, and the tiles go round it. You win when the numbers are in order, left to right and top to bottom, with the gap at the end.
Pocket Games offers three sizes: 4×4 (15 tiles, the classic), 5×5 (24 tiles) and 6×6 (35 tiles). Your best move count is recorded for each, so the game is not getting to the end: it is getting there in fewer moves than last time. On a computer you can play with the arrow keys.
One detail that matters: the starting grid is not random, it is the solved grid scrambled with legal moves. So every game is solvable. The physical puzzle is different: half of all possible arrangements have no solution. Sam Loyd built his fame on that, offering a thousand dollars for a position he knew was impossible — except the puzzle was not his, he claimed authorship years later.
The method: top to bottom, left to right
Everyone’s mistake is shoving tiles around until the mess moves somewhere else. The way through is different: finish one row and never touch it again.
1. The first row. Put 1, 2 and 3 in place, then deal with the 4. When you move on to the next row, pretend the first one does not exist: every manoeuvre from then on happens below it.
2. The middle rows. Same thing, always working only in the unfinished part, which gets shorter each time.
3. When two rows are left, change direction: solve them by columns, two tiles at a time, from the left. The last two rows can no longer be handled one at a time, because finishing the last row would break the one above it.
4. The final two tiles go in with one full turn of the gap around the closing 2×2 square.
The last-pair manoeuvre
This is where almost everyone gives up. Take the first row of a 4×4: the 4 belongs at the far right, but the moment you put it there, moving the 3 into place pushes it away. The answer is not to place them one at a time:
- Put the 3 where the 4 should go, at the end of the row.
- Put the 4 directly below the 3.
- Now rotate the pair: slide the 3 left and bring the 4 up. Both land in one go, and the row is closed.
The exact same shape works for the last tile of every row and, rotated ninety degrees, for the last tile of every column. Learn it once and you can close any grid, at any size.
Advice
- Move the gap, not the tiles. Thinking in terms of "where do I take the hole" makes obvious a lot of manoeuvres that otherwise look like magic.
- Only count moves from your second game. The first one is for learning the method; you improve your record afterwards, by cutting out the gap’s wasted loops.
- The 6×6 changes nothing. It is the same method with more rows: if it works on a 4×4 it works at any size, which is why it is worth learning instead of going on instinct.
Frequently asked questions
Can a grid be impossible?
In the physical puzzle yes, in this version no: since it is scrambled from the solution, you can always walk it back.
What is the minimum number of moves?
It depends on the scramble. For a 4×4 the worst possible case takes 80 moves; a normal game runs between sixty and a hundred and twenty, and getting under a hundred with the row method is a good target.
Why is it called the 15 puzzle?
Because in the classic version there are fifteen numbered tiles and the sixteenth square is empty. The 5×5 and 6×6 versions are the same puzzle, scaled up.
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